RUI: Existence and Stability of Coherent Structures with Applications to Elasticity

弹性杆理论的应用范围从模拟DNA的动力学到海洋电缆的盘绕。 常用的弹性杆动力学描述方程是由Kirchhoff提出的。 然而，基尔霍夫模型是复杂的，只有少数几类显式解是已知的和分析。 在一系列的论文中，Goriely和他的同事们得到了一个简化模型，该模型由超过第一次扭动不稳定性阈值的扭杆动力学的振幅方程组成。已知这些方程的解对应于杆的全三维变形。 该项目的目标是双重的。 首先，研究了一类振幅方程显式解的存在性和稳定性。 其次，将研究这些解的稳定性的方法推广到完全可积或具有一定对称性的偏微分方程。 更准确地说，第一个任务涉及振幅方程相干结构的存在性和稳定性的系统研究。 存在性研究的对称性分析的手段，而稳定性是使用两种不同的工具：埃文斯函数和哈密顿技术。第二个任务是对Evans函数的一般性研究。 研究者研究了埃文斯函数和完全可积性之间的关系，特别是对可积性的一个特殊概念--Painleve检验--的联系感兴趣。 当可积性不存在时，方程仍然可能具有有趣的李对称性。我们的目标是表明，重要的信息埃文斯功能可以从这些对称性检索。 力学是物理学中一个迷人的领域。 它的主要目的是研究粒子在引力等力存在下的相互作用。 与粒子动力学相反，连续介质（如流体或弹性体）的研究具有挑战性，而且尚未完全理解。 这是由于连续媒体以多种复杂的方式传播的事实。 例如，很难用数学描述弹性杆的扭曲和弯曲，但近似模型确实存在。 弹性杆的动力学研究是一个有趣的研究领域，因为它产生了美丽而复杂的数学结构。此外，了解弹性杆在科学的几个领域的应用是至关重要的。 在生物学中，杆状模型被用来描述不同结构的螺旋，如DNA。 在工程中，通常采用弹性杆来研究海底电缆的力学行为。 在这个项目中，该项目的重点是一个模型，已被证明是成功的描述棒的动力学：基尔霍夫方程。 这些是微分方程，也就是说，它们是包含导数的方程。 这些方程的解代表了弹性杆的可能行为。 本课题的主要目的是研究弹性杆模型的稳定性。 稳定性是物理学中的一个基本概念：例如，在理论上，可以让铅笔站在它的铅芯上，但在实践中，因为这是一个如此不稳定的状态，它不能做到。 在刚刚描述的例子中，稳定性的研究非常简单，不需要数学分析来证明或否定系统的稳定性。 稳定性的概念延续到微分方程的解，如基尔霍夫方程。 在这种情况下，稳定性采取了一种抽象的形式，它的研究往往涉及复杂的数学工具。 稳定性的研究是非常重要的，因为只有稳定的解才能通过实验得到。 不稳定的解，就像铅笔在笔尖上保持平衡的情况一样，虽然它们在理论上存在，但不会被观察到。 因此，通过稳定性研究，人们可以区分实验上可以实现的解决方案和不能实现的解决方案。 研究者提出了新的方法来研究数学稳定性。 他开发的方法被应用于弹性杆和其他应用。

Applications of elastic rod theory range from modeling thedynamics of DNA to the coiling of oceanic cables. The commonlyused descriptive equations for elastic rods dynamics are due toKirchhoff. However, the Kirchhoff model is complex and only fewclasses of explicit solutions are known and analyzed. In a seriesof papers, Goriely and coworkers obtained a reduced modelconsisting of amplitude equations for the dynamics of a twistedrod beyond the threshold of its first writhing instability. Solutions to these equations are known to correspond to fullythree-dimensional deformations of rods. The goals of this projectare twofold. Firstly, the investigator studies existence andstability properties for a large class of explicit solutions ofthe amplitude equations. Secondly, the methods developed to studythe stability of these solutions are extended to partialdifferential equations that are completely integrable or havecertain symmetry properties. More precisely, the first taskinvolves a systematic study of existence and stability of coherentstructures for amplitude equations. Existence is studied by meansof symmetry analysis, while stability is addressed using twodifferent tools: the Evans function and Hamiltonian techniques. The second task is a general study of the Evans function. Theinvestigator examines the relation between the Evans function andcomplete integrability and, in particular, is interested in theconnections with a particular notion of integrability, thePainleve Test. When integrability is not present, it is stillpossible for the equations to admit interesting Lie symmetries. The goal is to show that important information on the Evansfunction can be retrieved from these symmetries. Mechanics is a fascinating field of physics. Its mainpurpose is the study of interactions of particles in the presenceof forces such as gravity. In contrast to the dynamics ofparticles, the study of continuous media, such as fluids orelastic bodies, is challenging and not yet entirely wellunderstood. This is due to the fact that continuous media behavein numerous and complex ways. For example, it is difficult todescribe mathematically the twisting and bending of elastic rods,but approximate models do exist. The study of the dynamics ofelastic rods is a field of research that is interesting in itselfas it gives rise to beautiful and complex mathematical structures. Furthermore, understanding elastic rods is crucial forapplications in several domains of science. In biology, rodmodels are used to describe the coiling of different structuressuch as DNA. In engineering, elastic rods are used to study thebehavior of submarine cables. In this project, the investigatorfocuses on a model that has been proven to be successful indescribing the dynamics of rods: the Kirchhoff equations. Theseare differential equations, that is they are equations involvingderivatives. The solutions to these equations represent possiblebehaviors for elastic rods. The main purpose of the project is tostudy stability properties of elastic rod models. Stability is afundamental concept in physics: for example, in theory, it ispossible to make a pencil stand on its lead but, in practice,because that is such an unstable state, it cannot be done. In theexample just described, the study of stability is very simple andthere is no need for a mathematical analysis to prove or disprovethe stability of the system. The concept of stability carriesover to solutions of differential equations such as the Kirchhoffequations. In this context, stability takes an abstract form andits study often involves sophisticated mathematical tools. Thestudy of stability is of fundamental importance because onlystable solutions can be realized experimentally. Unstablesolutions, just like in the case of the pencil balancing on itspoint, although they exist in theory will not be observed. Hence,with stability studies, one can distinguish the solutions that canbe realized experimentally from the ones that cannot. Theinvestigator proposes new ways of investigating stabilitymathematically. The methods he develops are applied to elasticrods and other applications.